Shooting Star blender render
Final result:



Particles settings:

In this example Particles Emitter is small IcoSphere following spiral Curve.



Brownian - will scatter particles.

Damp - will keep them together.

Gravity - well, will make them less influenced by gravity :)

IcoSphere.001 - is our Particle Object.

Try them out to get desired result. 
(Constant rewinding video on YouTube is not my strong side.)

Particle Object material:



To the Particle Object (in this case IcoSphere.001) you will want to assign Particle Info node and change color of them by their Age, or read more here: How do I get particles to change color over time?

Math node > Divide allows you to change how quickly this change will be and ColorRamp gives you colors.

Blend file: